Rational BV-algebra in string topology

Abstract

Abstract. -Let M be a 1-connected closed manifold of dimension m and LM be the space of free loops on M . M. Chas and D. Sullivan defined a structure of BValgebra on the singular homology of LM , H * (LM ; k). When the ring of coefficients is a field of characteristic zero, we prove that there exists a BV-algebra structure on the Hochschild cohomology HH * (C * (M ); C * (M )) which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between HH * (C * (M ); C * (M )) and the shifted homology H * +m (LM ; k). We also prove that the Chas-Sullivan product and the BV-operator behave well with a Hodge decomposition of H * (LM ).Texte reçu le 7 juin

The Batalin-Vilkovisky Algebra on Hochschild Cohomology Induced by Infinity Inner Products

“…Thus, we can ask, if the BValgebra on H • (A, A) from theorem 2, and the one on H • (LX) from string topology coincide under this isomorphism. In fact, it has been shown by Felix-Thomas [6] and Merkulov [17], that in characteristic zero, the BValgebras on H • (A, A) and H • (LX) do coincide. In characteristic non-zero, this does not hold in general, when using the homological Poincaré duality pairing, cf.…”

Batalin-Vilkovisky algebra structures on Hochschild cohomology

“…In [29] Over Q, we explain in Corollary 20 how to put a Batalin-Vilkovisky algebra structure on HH * (S * (M ; Q), S * (M ; Q)) from a slight generalisation of Corollary 19 (Theorem 18). In fact both Félix, Thomas [12] and Chen [3,Theorem 5.4] proved that the Chas-Sullivan Batalin-Vilkovisky algebra H * +d (LM ; Q) is isomorphic to the Batalin-Vilkovisky algebra given by Corollary 20. Remark that, over Q, when the manifold M is formal, a consequence of Félix and Thomas work [12], is that H * +d (LM ) is always isomorphic to the BatalinVilkovisky algebra HH * (H * (M ); H * (M )) given by Corollary 19 applied to the symmetric algebra H * (M ). Over F 2 , in [24], we showed that this is not the case.…”

“…In [12] and [3,Theorem 5.4], it is shown that the Batalin-Vilkovisky algebra H p+d (LM ; Q) of Chas and Sullivan is isomorphic to the Batalin-Vilkovisky algebra on…”

On the loop homology of complex projective spaces

“…-In [6], the authors proved that rationaly for 1-connected closed oriented manifolds, there is an isomorphism of BV algebra betwen the Hochschild cohomology HH * (C * (M, Q), C * (M, Q)) and …”